A person 150cm tall is walking away from a lamp post at the rate of 15 meter per minute. when the man is 2.5m from the lamp post, his shadow is 3m long. Find the rate at which the length of the shadow is increasing when he is 7m from the lamp post.
when the man is x from the pole, and his shadow is s,
150/300 = h/(250+300)
so, the pole's height is h=275 cm
so, at any time,
150/s = 275/(x+s)
or, more easily manipulated,
(x+s)/275 = s/150
x/275 = s/330
x = 5/6 s
so, at any time
dx/dt = 5/6 ds/dt
since dx/dt = 15,
ds/dt = 18 m/s regardless of the distance from the pole.
To solve this problem, we can use similar triangles and the concept of related rates.
Let's define some variables:
- Let h be the height of the person (150 cm or 1.5 m)
- Let x be the distance between the person and the lamp post
- Let y be the length of the person's shadow
Given information:
- The person is walking away from the lamp post at a rate of 15 meters per minute.
- When the person is 2.5 m from the lamp post, his shadow is 3 m long.
First, let's express the relationship between x, y, and h based on similar triangles.
Using similar triangles, we know that the ratio of the person's height to the length of the shadow is constant.
Therefore, we can set up the following proportion:
x / y = h / (h + y)
Now, we need to find the rate at which the length of the shadow is increasing when the person is 7 m from the lamp post.
Let's find an expression for dy/dt (the rate at which the length of the shadow is changing with respect to time):
Differentiating implicitly with respect to t, we get:
(d/dt)(x/y) = (d/dt)(h / (h + y))
To find dx/dt, we need to use the given information that the person is walking away from the lamp post at a rate of 15 meters per minute:
dx/dt = -15 m/min (negative because the person is moving away from the lamp post)
Now, let's substitute the known values into the equation and solve for dy/dt:
-15 / y = 1.5 / (1.5 + y)
Simplifying the equation, we get:
-15y = 1.5(1.5 + y)
-15y = 2.25 + 1.5y
Combine like terms:
-15y - 1.5y = 2.25
-16.5y = 2.25
Dividing by -16.5, we get:
y = -2.25 / 16.5
y ≈ -0.1364
Since length cannot be negative, the negative sign is due to the direction. We can therefore take the positive value of y:
y ≈ 0.1364
Now, let's find the rate at which the length of the shadow is increasing when the person is 7 m from the lamp post.
To do this, we substitute the given values into the equation and solve for dy/dt:
-15 / 7 = 1.5 / (1.5 + 0.1364) * dy/dt
-2.1429 = 1.5 / 1.6364 * dy/dt
-2.1429 ≈ 0.9174 * dy/dt
Dividing by 0.9174, we get:
dy/dt ≈ -2.1429 / 0.9174
dy/dt ≈ -2.3415
The rate at which the length of the shadow is increasing when the person is 7 m from the lamp post is approximately -2.3415 meters per minute. The negative sign indicates that the shadow is getting shorter.